Real-rooted atlas

A391045 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 - 6\,x - 6\,x^{2} - 2\,x^{3} + 8\,n + 24\,n x + 24\,n x^{2} + 8\,n x^{3} - 10\,n^{2} - 30\,n^{2} x - 30\,n^{2} x^{2} - 10\,n^{2} x^{3} + 4\,n^{3} + 12\,n^{3} x + 12\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n}(x) = \left(-2 - 44\,x - 210\,x^{2} - 440\,x^{3} - 470\,x^{4} - 252\,x^{5} - 54\,x^{6} + 8\,n + 140\,n x + 613\,n x^{2} + 1214\,n x^{3} + 1244\,n x^{4} + 646\,n x^{5} + 135\,n x^{6} - 10\,n^{2} - 138\,n^{2} x - 530\,n^{2} x^{2} - 946\,n^{2} x^{3} - 888\,n^{2} x^{4} - 428\,n^{2} x^{5} - 84\,n^{2} x^{6} + 4\,n^{3} + 44\,n^{3} x + 147\,n^{3} x^{2} + 232\,n^{3} x^{3} + 194\,n^{3} x^{4} + 84\,n^{3} x^{5} + 15\,n^{3} x^{6}\right)\,P_{n-1}(x) + \left(8\,x - 4\,x^{2} - 150\,x^{3} - 400\,x^{4} - 460\,x^{5} - 252\,x^{6} - 54\,x^{7} - 18\,n x + 8\,n x^{2} + 320\,n x^{3} + 840\,n x^{4} + 950\,n x^{5} + 512\,n x^{6} + 108\,n x^{7} + 8\,n^{2} x - 2\,n^{2} x^{2} - 118\,n^{2} x^{3} - 292\,n^{2} x^{4} - 308\,n^{2} x^{5} - 154\,n^{2} x^{6} - 30\,n^{2} x^{7}\right)\,P_{n-1}'(x) + \left(3\,x^{2} + 6\,x^{3} - 15\,x^{4} - 60\,x^{5} - 75\,x^{6} - 42\,x^{7} - 9\,x^{8} - 5\,n x^{2} - 10\,n x^{3} + 25\,n x^{4} + 100\,n x^{5} + 125\,n x^{6} + 70\,n x^{7} + 15\,n x^{8}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2134
31101223
41216953144
5136228432244941
61555701950258011526308
7178120064401477514982553143007
81105224717395607251040138540726875296768
911363864407682006205117286969484805521317602066377
101171622885932566244200062839943684507344267771665049214489124
111210954016680014177106612228180429902949444028383120148090403229368102165447
12125314025303105322905019200258683792341511812502088088201750542208141328816105344723688848
13130019932521840681120050276160226076004646421688119754805514302818201061823642445401948806249355145807525
141351275348588581348204012095226066963468023983330805703521967907522258595401125826353427522242692906340491322536707415748
15140637128136063225294269271080810181127446579316094002347065220547645988390663550507626225806059237585736461131790919542039146908262574384383
Technical details
Node id
n-9347baaf590c8ca8 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-bc84d4d7514c6de4

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   3;
   3: 1,  10,   12;
   4: 1,  21,   69,    53;
   5: 1,  36,  228,   432,    244;
   6: 1,  55,  570,  1950,   2580,    1152;
   7: 1,  78, 1200,  6440,  14775,   14982,    5531;
   8: 1, 105, 2247, 17395,  60725,  104013,   85407,   26875;
   9: 1, 136, 3864, 40768, 200620,  511728,  696948,  480552,  131760;
  10: 1, 171, 6228, 85932, 566244, 2000628, 3994368, 4507344, 2677716, 650492;
%F Formula
The row polynomials satisfy (4*n^3-10*n^2+8*n-2+(12*n^3-30*n^2+24*n-6)*x+(12*n^3-30*n^2+24*n-6)*x^2+(4*n^3-10*n^2+8*n-2)*x^3)*P(n,x) = (4*n^3-10*n^2+8*n-2+(44*n^3-138*n^2+140*n-44)*x+(147*n^3-530*n^2+613*n-210)*x^2+(232*n^3-946*n^2+1214*n-440)*x^3+(194*n^3-888*n^2+1244*n-470)*x^4+(84*n^3-428*n^2+646*n-252)*x^5+(15*n^3-84*n^2+135*n-54)*x^6)*P(n-1,x) + ((8*n^2-18*n+8)*x-(2*n^2-8*n+4)*x^2-(118*n^2-320*n+150)*x^3-(292*n^2-840*n+400)*x^4-(308*n^2-950*n+460)*x^5-(154*n^2-512*n+252)*x^6-(30*n^2-108*n+54)*x^7)*d/dx P(n-1,x) - ((5*n-3)*x^2+(10*n-6)*x^3-(25*n-15)*x^4-(100*n-60)*x^5-(125*n-75)*x^6-(70*n-42)*x^7-(15*n-9)*x^8)*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^3 - 10*n^2 + 8*n - 2 + (44*n^3 - 138*n^2 + 140*n - 44)*x + (147*n^3 - 530*n^2 + 613*n - 210)*x^2 + (232*n^3 - 946*n^2 + 1214*n - 440)*x^3 + (194*n^3 - 888*n^2 + 1244*n - 470)*x^4 + (84*n^3 - 428*n^2 + 646*n - 252)*x^5 + (15*n^3 - 84*n^2 + 135*n - 54)*x^6)*P[n - 1, x] + ((8*n^2 - 18*n + 8)*x - (2*n^2 - 8*n + 4)*x^2 - (118*n^2 - 320*n + 150)*x^3 - (292*n^2 - 840*n + 400)*x^4 - (308*n^2 - 950*n + 460)*x^5 - (154*n^2 - 512*n + 252)*x^6 - (30*n^2 - 108*n + 54)*x^7)*D[P[n - 1, x], x] - ((5*n - 3)*x^2 + (10*n - 6)*x^3 - (25*n - 15)*x^4 - (100*n - 60)*x^5 - (125*n - 75)*x^6 - (70*n - 42)*x^7 - (15*n - 9)*x^8)*D[P[n - 1, x], {x, 2}])/(4*n^3 - 10*n^2 + 8*n - 2 + (12*n^3 - 30*n^2 + 24*n - 6)*x + (12*n^3 - 30*n^2 + 24*n - 6)*x^2 + (4*n^3 - 10*n^2 + 8*n - 2)*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4*n^3 - 10*n^2 + 8*n - 2 + (44*n^3 - 138*n^2 + 140*n - 44)*x + (147*n^3 - 530*n^2 + 613*n - 210)*x^2 + (232*n^3 - 946*n^2 + 1214*n - 440)*x^3 + (194*n^3 - 888*n^2 + 1244*n - 470)*x^4 + (84*n^3 - 428*n^2 + 646*n - 252)*x^5 + (15*n^3 - 84*n^2 + 135*n - 54)*x^6)*v[i - 1] + ((8*n^2 - 18*n + 8)*x - (2*n^2 - 8*n + 4)*x^2 - (118*n^2 - 320*n + 150)*x^3 - (292*n^2 - 840*n + 400)*x^4 - (308*n^2 - 950*n + 460)*x^5 - (154*n^2 - 512*n + 252)*x^6 - (30*n^2 - 108*n + 54)*x^7)*deriv(v[i - 1], x) - ((5*n - 3)*x^2 + (10*n - 6)*x^3 - (25*n - 15)*x^4 - (100*n - 60)*x^5 - (125*n - 75)*x^6 - (70*n - 42)*x^7 - (15*n - 9)*x^8)*deriv(deriv(v[i - 1], x), x))/(4*n^3 - 10*n^2 + 8*n - 2 + (12*n^3 - 30*n^2 + 24*n - 6)*x + (12*n^3 - 30*n^2 + 24*n - 6)*x^2 + (4*n^3 - 10*n^2 + 8*n - 2)*x^3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023